This course is an introduction to the finite element method as applicable to a range of problems in physics and engineering sciences. The treatment is mathematical, but only for the purpose of clarifying the formulation. The emphasis is on coding up the formulations in a modern, open-source environment that can be expanded to other applications, subsequently. The course includes about 45 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because the main goal of these lectures is to turn the viewer into a competent developer of finite element code. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed. The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based. Books: There are many books on finite element methods. This class does not have a required textbook. However, we do recommend the following books for more detailed and broader treatments than can be provided in any form of class: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes, Dover Publications, 2000. The Finite Element Method: Its Basis and Fundamentals, O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, Butterworth-Heinemann, 2005. A First Course in Finite Elements, J. Fish and T. Belytschko, Wiley, 2007. Resources: You can download the deal.ii library at dealii.org. The lectures include coding tutorials where we list other resources that you can use if you are unable to install deal.ii on your own computer. You will need cmake to run deal.ii. It is available at cmake.org.
01.03. Boundary conditions
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The Finite Element Method for Problems in Physics
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This unit is an introduction to a simple one-dimensional problem that can be solved by the finite element method.
Rencontrer les enseignants
Krishna Garikipati, Ph.D.Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts
>> Welcome back.
We'll continue with our development of, our one dimensional elliptic
partial differential equation and we have been doing it so far using elasticity
as a canonical physical problem, right, to guide us through this process.
So we had got as far as writing out the differential equation and
we were talking about boundary conditions.
So let's continue talking about, about boundary conditions, okay?
So let me actually draw our domain once again.
We have the bar.
We have our coordinate x.
We have x equals 0 and we have x equals L.
Now, At the left end, x equals 0.
We have u 0 equals u naught.
Right? Specified as a condition.
And at the right end, we have either
u at L equals u sub g for given,
or we talked about how we have
sigma at L equals t, okay?
And at the very end of the previous segment, we had talked about
you know, the nomenclature that we employ for these different boundary conditions.
Now I want to point out something that's actually quite important both from
a mathematical and a physical standpoint for this problem.
Which is the following, observe that we always have
this the Dirichlet boundary condition at x equals 0.
However, at x equals L, we have either
a Dirichlet boundary condition or a Neumann boundary condition.
Okay?
In, in particular we do not have a situation where
at both ends we may have Neumann boundary conditions.
Okay. So let me state that.
Okay.
So in terms of boundary conditions
>> Okay?
We have either
>> case a, being Dirichlet
boundary conditions.
As we go on, I'll be offering you as BCS short for boundary conditions.
So either way, we have Dirichlet
boundary conditions at x equals 0 and
x equals L, right?
Or b.
Dirichlet boundary
condition at x equals 0 and
Neumann boundary
condition at x equals L.
It turns out that in case b we, we could actually of flipped things around.
Which is that we could have had Dirichlet boundary condition x equals l,
and a Neumann boundary condition at x equals zero that would not pose a problem.
Okay, what's important is to note
that there is a particular combination that we are not considering.
Okay?
It's very obvious but I'll let you think about it for a second or
two, to make sure that you are indeed with me.
This.
Okay.
The condition we are not considering is Neumann at both ends.
Okay?
So, we do not
consider Neumann
boundary conditions
at x equals zero and
x equals L.
All right?
There's a very good reason for this.
Okay?
Let me try and motivate the reason first from a,
from a physical standpoint, okay?
What will, why this is important is the following, is, is, is the following.
For the kind of problem we are considering,
we're saying that we always need to have the displacement specified in one end or
the other, at least at one end, okay?
So we may have the dis, the displacement specified at this end of where I'm holding
it and on, on the other end we may either specify the displacement or a force.
That's okay.
Right? Or attraction, sorry.
That's okay.
Okay?
Or what we are not allowed to do is to say that we
have the traction specified at one end as well at, as, as at the other.
Okay? So we can't have Neumann
boundary conditions on both sides, okay?
And now you may wonder why this is the case.
In particular you may ask the question, but
surely that leaves out a very important class of problems.
In particular, that does leave, it seems to leave out the class of problems where
I mean I'll take this bar, right?
Remember it's after all an, an elasticity problem, right.
You may say that it, we're leaving out the pro,
the class of problems where we take this bar and sort of throw it one way.
Remember we're doing it one way, so
we're assuming that it moves only one direction along the x direction.
If we, if we, if we toss this bar and it moves along the x direction,
we may say well, we've ruled out the ability to solve that problem because
we have stated, so far in our development,
that we always have the displacement specified at one end or the other.
Okay? If we were to toss it and
just watch it, watch it evolve in time,
we would not be specifying the displacement at one end, right?
It would just be, we would be throwing it and let it evolve in, in space and
time, but we would not have specified the displacement at one end.
Okay?
The reason is that, the reason we,
we, we have this restriction is because we are considering steady state problems.
The problem where we actually throw the bar and watch it evolve in, in,
in space and time would be a time dependent problem.
Okay?
And in particular in the context of elasticity it would no longer be
an elliptic problem, it would be a hyperbolic problem.
There are other ways in which that problem is treated.
In particular, we would need to specify an additional set of conditions.
Can you think of what those may be?
We would need initial conditions as well.
Okay?
What does that mean mathematically, 'kay?
What it means mathematically is the following, okay?
The reason we do not consider Neuman boundary conditions in,
in this description of the problem is because of the following, okay?
Supposing you consider,
let, let's suppose we do try to specify non boundary conditions at both ends.
Okay?
So consider, Neumann
boundary conditions at x equals 0 and
x equals L, okay?
So what this may mean is that we are saying
that sigma at 0 which we know to be E u,x
evaluated at x equals 0 equals let's say.
T, 0 right for traction at zero?
And sigma at L which is what we have indeed considered is E times u,
x evaluated at x equals L now equal to T sub L.
Okay? I'm distinguishing T0, T not and
T L just by the positions, okay?
Now you observe that this
poses a certain problem because supposing so, so if you had these two conditions
Our, problem would be one of finding this basement field which,
in addition to satisfying these two boundary conditions also, of course,
needs to satisfy the differential equation.
Right?
So we had this and
now we say consider U of
X satisfying these Neumann
boundary conditions and
the differential equation.
Right and the differential equation
remember is the sigma d x plus f equals 0.
Right?
In 0, L.
Okay.
Now, if you had this sort of situation,
making our substitution come from the constitutive relation tells us that now
d dx of E times u,x,
where I've simply substituted our constitutive relation for this stress,
plus f equals 0, okay?
Now.
This is the problem we would be solving, right?
We, we would be solving a problem with these two boundary conditions and
this, differential equation.
Now observe that any solution we get to this problem would
be non unique up to a constant displacement field.
Okay?
Because supposing you did have a u that satisfied these conditions.
Right?
Alright.
What we could do is that u of x plus
some other u which I will denote as u bar.
Okay?
Which is a constant.
With respect to wrt being short for with respect to.
X, okay?
This field is also a solution.
Right?
Clearly if u, if u bar is a constant with respect to x,
it would satisfy our two Neumann boundary conditions.
It would satisfy that.
And it would satisfy that because derivatives with respect to u, sorry,
derivatives with respect to x of u bar would be zero.
It would not contribute to that boundary condition.
Likewise to the differential equation, right?
Because the differential equation actually involves two
derivatives on the displacement field.
Right?
If, if, if E were a constant.
Or even if E were not a constant it would involve at least one derivative on,
on, on the displacement field.
Right? So therefore
adding on U bar equals constant would still satisfy that, that equation as well.
Okay?
So what we see is that If we had Neumann the boundary
conditions only on our problem the solution
u of x is non unique.
[SOUND] Up to a constant displacement field and
in the context of elasticity, a constant
displacement field is also called a rigid body motion.
Right? Because remember,
this is our one dimensional domain over which we're trying to solve this problem.
What is a displacement field that is constant with respect to position?
It's one in which we seeing,
we're seeing that the entire bar has a translation, right?
Every point in the bar has the same displacement,
u bar, added on to any solution we specify to it.
Okay?
So we say that the solution
u x is non-unique up to
a rigid body motion.
U bar, okay,
which is
a constant.
Okay?
In this particular say, case, saying a rigid body motion and
saying u bar is a constant are actually superfluous.
They both mean the same thing.
Okay?
So it is that for these kinds of problems that we are looking at, in particular
elliptic differential equations in one dimension, okay?
We cannot have Neumann boundary conditions, only.
We do need a traditionally boundary condition and it is
exactly to make sure that our solution is unique with respect to the solution,
with respect to the primary field.
Okay?
And also remember that I said that,
that, we did talk about how the problem of actually tossing this bar and
watching it move in space and time is a different problem.
Okay it's no longer an elliptic problem.
In particular, it's no longer a steady state problem.
Because we would be looking at how a solution evolves in time by
not assuming that things are constant In time.
Okay.
So maybe I should state that as well.
Neumann boundary conditions
alone can be specified.
For the time-dependent.
Elasticity problem.
Okay?
And I'll just state here that though we won't be
coming back to this time-dependent elasticity problem.
I will just state for
now that this would then become what is called a hyperbolic differential equation.
The hyperbolic partial differential equation.
Okay?
We are not doing that right now.
We are looking at elliptic partial differential equations, which for
the kinds of things we are looking at also means that we have steady state
problems, okay?
So those are the important things we need to say about the boundary conditions.
What I will also do now is spend a few minutes talking about
what I've been calling our body force, okay?
So let us now recall.
The body force that we've written as f has a function of x.
Okay, and the role it plays in our problem is the following.
It gives us d sigma dx plus f, and
here I'll make it explicit that it depends upon position, is equal to 0 in.
Okay?
In this case, what I'm calling the body force,
of course is a term that is applicable only when we're doing elasticity.
Okay?
In more general problems,
in more general PTEs, this would just be called the forcing function.
Okay?
So this is the.
Is the force and function in other pdes
probably not other pdes.
Maybe I should just call this in general pdes, all right,
simply because the term body force may not have any significance in other pdes.
All right.
It's just, it's just the forcing function.
Nevertheless, since we are talking in the context of,
we are talking of elasticity let me state what this forcing function is.
Okay.
Give you, give you an example of the forcing function.
So, as an example, let's suppose that we have again this sort of situation.
We have the bar.
Kay.
It extends along the x direction.
0 and l, and that, I had this sort of picture that I drew early on, right?
Just say that that is our distribution of the body force, right?
F.
So a specific example of this would be the following.
Supposing that although I've drawn this body as extending in the horizontal
direction at least relative to the, this
particular screen let's also suppose that, interestingly enough in this situation,
this was the direction of the action of the acceleration due to gravity.
Okay? I just happened to turn my
coordinate system around so that it turned out to be horizontal but that's the,
the, the direction of gravity.
Okay?
So G is D.
Acceleration- Due
to- Gravity, okay?
In that situation, f of x
would be the mass density of the body, which, you know,
could be a function of position, right?
Maybe this bar does not have a constant density with respect to position times g.
So is this a specific example of the body force.
And know, perhaps make it a little more physical maybe we should
say that this bar, you know, is actually turned around this way.
So, downward is the x direction, and
now you can see very clearly how that would be the body force, right?
It could simply be the the distributed force.
Distributed because it's distributed at every point in the, every, every,
every, every little mass element of our bar.
And it's the force that's distributed in every mass element
of our bar due to gravity.
Okay, so that's another example of it.
Okay.
So, I think we can stop this segment here.
When we come back,
we will continue with, saying a little more about this, differentiation.
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